The derivative nonlinear Schr\"odinger (DNLS) equation is the canonical model for dynamics of waves in plasma physics and optics. We study exact solutions describing rogue on background periodic standing DNLS equation. show that space-time localization a wave only possible if modulationally unstable. If stable, degenerate into algebraic solitons propagating along interacting with waves. Maximal...

The Talbot effect is usually modeled using the Helmholtz equation, but its main experimental features are captured by solution to free Schr\"odinger equation with Dirac comb as initial datum. This simplified description a consequence of paraxial approximation in geometric optics. However, it heuristic that not mathematically well justified, so K. I. Oskolkov raised problem "mathematizing" it. W...

The dynamics of vortex laments has provided for almost a century one of the most interesting connections between diierential geometry and integrable equations, and an example in which knotted curves arise as solutions of diierential equations possessing remarkably rich geometrical structures. In this paper we present several aspects of the integrable evolution of a closed vor-tex lament in an E...

For the magnetic Hamiltonian with singular vector potentials, we analytically continue resolvent to a logarithmic neighborhood of positive real axis and prove estimates there. As applications, obtain asymptotic locations resonances local smoothing estimate for solutions corresponding Schr\"odinger equation.

Recent advances in the treatment of scattering of charged composite particles are reviewed. In a rst part I report on developments of the theory. Speciically I describe the recent completion of the derivation of the coordinate space asymptotic behaviour of the wave function for three charged particles in the continuum. This knowledge is increasingly being made use of in attempts tòderive' three...

This article reviews theoretical methods to deal with interacting quantum particles that are in contact their environment and thus described by a master equation rather than Schr\"odinger equation. The similarities differences discussed between the pursuit of pure many-body ground states mixed steady different methods, an outlook is provided on advances toward simulation large open system.

In this paper, we prove that the cubic nonlinear Schr\"odinger equation with fractional Laplacian on unit disk is globally well-posed for certain radial initial data below energy space. The result proved by extending I-method in setting.

We consider two dimensional Grushin Schr\"odinger equation posed on a finite cylinder $\Omega=(-1,1)_x\times \T_y$ with Dirichlet boundary condition. obtain the sharp observability by any horizontal strip, optimal time $T_*>0$ depending size of strip. Consequently, we prove exact controllability for equation. By exploiting concentration eigenfunctions harmonic oscillator at $x=0$, also show tha...

We investigate the solutions for a time dependent potential by considering two scenarios fractional Schr\"odinger equation. The first scenario analyzes influence of in absence kinetic term. obtain analytical and numerical this case Caputo derivative, which extends Rabi's model. In second scenario, we incorporate term equation consider spatial derivatives. For case, analyze spreading Gaussian wa...